Existence of positive solution for singular fractional differential equation

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error.

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The Iterative Scheme and the Convergence Analysis of Unique Solution for a Singular Fractional Differential Equation from the Eco-Economic Complex System’s Co-Evolution Process

Complexity ◽

10.1155/2019/9278056 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15 ◽

Cited By ~ 11

Author(s):

Teng Ren ◽

Helu Xiao ◽

Zhongbao Zhou ◽

Xinguang Zhang ◽

Lining Xing ◽

...

Keyword(s):

Differential Equation ◽

Convergence Analysis ◽

Unique Solution ◽

Fractional Differential Equation ◽

Iterative Process ◽

Iterative Scheme ◽

Singular Fractional Differential Equation ◽

Complex Processes ◽

Fractional Differential ◽

And Diffusion

In this paper, we focus on a class of singular fractional differential equation, which arises from many complex processes such as the phenomenon and diffusion interaction of the ecological-economic-social complex system. By means of the iterative technique, the uniqueness and nonexistence results of positive solutions are established under the condition concerning the spectral radius of the relevant linear operator. In addition, the iterative scheme that converges to the unique solution is constructed without request of any monotonicity, and the convergence analysis and error estimate of unique solution are obtained. The numerical example and simulation are also given to demonstrate the application of the main results and the effectiveness of iterative process.

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Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

Discrete Dynamics in Nature and Society ◽

10.1155/2015/736108 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

Jun-Rui Yue ◽

Jian-Ping Sun ◽

Shuqin Zhang

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Value Problem ◽

Fractional Derivative ◽

Positive Solution ◽

Fractional Differential Equation ◽

Nonlinear Fractional Differential Equation ◽

Krasnoselskii Fixed Point Theorem ◽

Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)), t∈[0,1], u(0)=0, u′(0)+u′′(0)=0, u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

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Existence and Uniqueness of Positive Solutions for Singular Fractional Differential Equation with Nonlocal Boundary Conditions

Pure Mathematics ◽

10.12677/pm.2020.103022 ◽

2020 ◽

Vol 10 (03) ◽

pp. 150-161

Author(s):

雪雪霍

Keyword(s):

Differential Equation ◽

Boundary Conditions ◽

Fractional Differential Equation ◽

Positive Solutions ◽

Existence And Uniqueness ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Singular Fractional Differential Equation ◽

Fractional Differential

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Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

Discrete Dynamics in Nature and Society ◽

10.1155/2012/425408 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 2

Author(s):

Changyou Wang ◽

Haiqiang Zhang ◽

Shu Wang

Keyword(s):

Differential Equation ◽

Positive Solution ◽

Fractional Differential Equation ◽

Caputo Derivative ◽

Nonlinear Term ◽

Upper And Lower Solutions ◽

Initial Value ◽

Nonlinear Fractional Differential Equation ◽

Control Functions ◽

Fractional Differential

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.

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